Math Problem Statement
Let f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column 6 minus x comma 2nd Column x less than 3 2nd Row 1st Column x plus 2 comma 2nd Column x greater than 3. EndMatrix
Complete parts a through d below. 0 5 10 0 5 10 x y y equals 6 minus xy equals x plus 2
A coordinate system has a horizontal x-axis labeled from 0 to 10 in increments of 1 and a vertical y-axis labeled from 0 to 10 in increments of 1. A graph has two branches. The first branch is a ray that falls from left to right, starting at (6,0) and ending at an open dot at (3,3). The first branch is labeled y equals 6 minus x. The second branch is a ray that rises from left to right, starting at an open dot at (3, 5) and passing through (4, 6). The second branch is labeled y equals x plus 2. Question content area bottom Part 1 a. Find ModifyingBelow lim With x right arrow 3 Superscript plus f left parenthesis x right parenthesis and ModifyingBelow lim With x right arrow 3 Superscript minus f left parenthesis x right parenthesis. Select the correct choice below and, if necessary, fill in any answer box(es) in your choice.
Solution
To solve the problem, we need to find the left-hand limit () and the right-hand limit () of the piecewise function as approaches 3.
The function is defined as:
\begin{cases} 6 - x, & \text{if } x < 3 \\ x + 2, & \text{if } x > 3 \end{cases}$$ ### Step 1: Find the Left-Hand Limit ($$\lim_{{x \to 3^-}} f(x)$$) The left-hand limit is the value that $$f(x)$$ approaches as $$x$$ gets closer to 3 from the left side ($$x < 3$$). For $$x < 3$$, the function is given by $$f(x) = 6 - x$$. So, $$\lim_{{x \to 3^-}} f(x) = 6 - 3 = 3$$ ### Step 2: Find the Right-Hand Limit ($$\lim_{{x \to 3^+}} f(x)$$) The right-hand limit is the value that $$f(x)$$ approaches as $$x$$ gets closer to 3 from the right side ($$x > 3$$). For $$x > 3$$, the function is given by $$f(x) = x + 2$$. So, $$\lim_{{x \to 3^+}} f(x) = 3 + 2 = 5$$ ### Conclusion - The left-hand limit is $$\lim_{{x \to 3^-}} f(x) = 3$$. - The right-hand limit is $$\lim_{{x \to 3^+}} f(x) = 5$$. Since the left-hand limit and the right-hand limit are not equal, the overall limit $$\lim_{{x \to 3}} f(x)$$ does not exist. ### Answer: The correct choice is: $$\lim_{{x \to 3^-}} f(x) = 3 \quad \text{and} \quad \lim_{{x \to 3^+}} f(x) = 5$$ Would you like to explore this further or see how the rest of the question is solved? Here are five related questions that might interest you: 1. How do you find the overall limit of a piecewise function when the left-hand and right-hand limits are different? 2. What is the graphical interpretation of a limit at a point where the function is discontinuous? 3. How does the concept of one-sided limits relate to continuity at a point? 4. How would you find the limit at a point where the function is continuous? 5. What are some common situations where limits do not exist? **Tip:** When analyzing piecewise functions, always consider the behavior of the function from both sides of the point in question to determine limits and continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Limits
One-sided limits
Formulas
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Theorems
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Suitable Grade Level
High school